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Journal of the London Mathematical Society 1981 s2-24(3):467-479; doi:10.1112/jlms/s2-24.3.467
© 1981 by London Mathematical Society
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© Oxford University Press

Hilbert Functions and Pseudo-Rational Local Rings of Dimension Two

D. Rees

Department of Mathematics, University of Exeter North Park Road, Exeter EX4 4QE

Let Q be an analytically unramified Cohen-Macaulay local ring of dimension 2, with maximal ideal m and infinite residue field k. If a is an m-primary ideal of Q, a* will denote its integral closure, {lambda}(a) = l(Q/a*), e(a) will denote its multiplicity, {theta}(a) = 2. {lambda}(a)–e(a) and Formula(a) = lim {theta}(an)/n. This paper uses explicit formulae for {theta}(ar), {theta}(arbs) related to earlier results of Narita to prove that Formula(ab) = Formula(a) + Formula(b), and that the identity {theta}(ab) = {theta}(a) + {theta}(b) holding for all m-primary ideals a, b characterises the pseudo-rational local rings of J. Lipman among normal Q. This is used to prove a number of results concerning the normal genus of an ideal and pseudo-rational local rings. In the last section, an expression for {theta}(a) is obtained in the case where Q is regular which is related to the theory of infinitely near points of D. G. Northcott.


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